Regularity and convergence of 4-dimensional extremal Kahler metrics

2011/04/16 by Brian Weber, Weber, Brian
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #math.DG

paper · pdf · doi:10.48550/arxiv.1104.3190

The paper has been temporarily withdrawn pending some corrections and improvements in format. In particular, some arguments surrounding volume comparison require clarification, and some parts of the paper may be broken off and released separately

arxiv created 2011/05/10 · arxiv updated 2011/05/11

Abstract

We establish a regularity result for the metric on any 4-dimensional extremal Kähler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity L²(|\Riem|) in a surrounding ball is sufficiently small compared to the pointwise norm of its scalar curvature. Consequently sequences of 4-dimensional extremal Kähler metrics with uniformly bounded Calabi energies and scalar curvature have convergent subsequences in the Gromov-Hausdorff topology. Gromov-Hausdorff limits are length spaces with the structure of Riemannian orbifolds away from finitely many point-like singularities of unknown structure.

Related

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